In this essay, I will explore a mathematical property of musical modes that may not have direct practical use for musicians or composers, but could intrigue mathematically inclined readers: the notion of symmetry. Let’s consider the C Ionian mode—also known as the C major scale. When played in ascending order, we can ask: what scale would produce the same sequence of intervals if played in descending order? Put differently, what is the symmetric scale—one that mirrors the original intervals in reverse? This is precisely what emerges when we apply the interval pattern in reverse, mode by mode. In the table below, intervals are measured in semitones (“s”).

Mode	Intervals	Reversed Intervals	Mode of Reversed Intervals
Ionian	2s 2s 1s 2s 2s 2s 1s	1s 2s 2s 2s 1s 2s 2s	Lydian
Dorian	2s 1s 2s 2s 2s 1s 2s	2s 1s 2s 2s 2s 1s 2s	Phrygian
Phrygian	1s 2s 2s 2s 1s 2s 2s	2s 2s 1s 2s 2s 2s 1s	Dorian
Lydian	2s 2s 2s 1s 2s 2s 1s	1s 2s 2s 1s 2s 2s 2s	Ionian
Mixolydian	2s 2s 1s 2s 2s 1s 2s	2s 1s 2s 2s 1s 2s 2s	Locrian
Aeolian	2s 1s 2s 2s 1s 2s 2s	2s 2s 1s 2s 2s 1s 2s	Aeolian
Locrian	1s 2s 2s 1s 2s 2s 2s	2s 2s 2s 1s 2s 2s 1s	Mixolydian

Let us now delve a little deeper in this topic. The question I would like to answer is whether the mode-symmetry tables represented above abide by a mathematical equation. In the following page, we will see that the answer is affirmative. For convenience’s sake, we will need to represent the seven modes with their respective ordinal.

Ordinal	Mode
1	Ionian
2	Dorian
3	Phrygian
4	Lydian
5	Mixolydian
6	Aeolian
7	Locrian

Proposition

Let us consider any given mode out of the list above. We will refer to it as mode(i), where i is its ordinal. The symmetric mode of mode(i) is mode(j), where i and j satisfy the following equation:

(i + j) mod 7 = 4

The proof will proceed by case:

• Ionian

As we have seen above, the symmetric mode of Ionian is Phrygian. Ionian has ordinal 1, whereas Phrygian has 3. Hence: (1 + 3) mod 7 = 4 mod 7 = 4

• Dorian

As we have seen above, the symmetric mode of Dorian is Dorian itself. Dorian has ordinal 2. Hence: (2 + 2) mod 7 = 4 mod 7 = 4

• Phrygian

Since symmetry is, by definition, a symmetric relation, the fact that Ionian is the symmetric mode of Phrygian implies that Phrygian is the symmetric mode of Ionian.

• Lydian

As we have seen above, the symmetric mode of Lydian is Locrian. Lydian has ordinal 4, whereas Locrian has 7. Hence: (4 + 7) mod 7 = 11 mod 7 = 4

• Mixolydian

As we have seen above, the symmetric mode of Mixolydian is Aeolian. Mixolydian has ordinal 5, whereas Aeolian has 6. Hence: (5 + 6) mod 7 = 11 mod 7 = 4

• Aeolian

Since symmetry is, by definition, a symmetric relation, the fact that Mixolydian is the symmetric mode of Aeolian implies that Aeolian is the symmetric mode of Mixolydian.

• Locrian

Since symmetry is, by definition, a symmetric relation, the fact that Lydian is the symmetric mode of Locrian implies that Locrian is the symmetric mode of Lydian.

♦